We have shown that for all real x, sin2 x + cos2 x = 1
It follows that
1 + tan2 x = sec2 x (why?)
1 + cot2 x = cosec2 x (why?)
In earlier classes, we have discussed the values of trigonometric ratios for 0°, 30°, 45°, 60° and 90°. The values of trigonometric functions for these angles are same as that of trigonometric ratios studied in earlier classes. Thus, we have the following table:
The values of cosec x, sec x and cot x are the reciprocal of the values of sin x, cos x and tan x, respectively.

Sign of trigonometric functionsLet P (a, b) be a point on the unit circle with centre at the origin such that ∠AOP = x. If ∠AOQ = – x, then the coordinates of the point Q will be (a, –b) (Fig 3.7).
Therefore
cos (– x) = cos x
and sin (– x) = – sin x
Since for every point P (a, b) on the unit circle, – 1 ≤ a ≤ 1 and – 1 ≤ b ≤ 1, we have – 1 ≤ cos x ≤ 1 and –1 ≤ sin x ≤ 1 for all x. We have learnt in previous classes that in the first quadrant (0 < x < π/2) a and b are both positive, in the second quadrant (π/2 < x < π) a is negative and b is positive, in the third quadrant (π < x < 3π/2 ) a and b are both negative and in the fourth quadrant (3π/2 < x < 2π) a is positive and b is negative. Therefore, sin x is positive for 0 < x < π, and negative for π < x < 2π. Similarly, cos x is positive for 0 < x < π/2, negative for π/2 < x < 3π/2 and also positive for 3π/2 < x < 2π. Likewise, we can find the signs of other trigonometric functions in different quadrants. In fact, we have the following table.

Domain and range of trigonometric functionsFrom the definition of sine and cosine functions, we observe that they are defined for all real numbers. Further, we observe that for each real number x,
– 1 ≤ sin x ≤ 1 and – 1 ≤ cos x ≤ 1
Thus, domain of y = sin x and y = cos x is the set of all real numbers and range is the interval [–1, 1], i.e., – 1 ≤ y ≤ 1.
Since cosec x = 1/sin x , the domain of y = cosec x is the set { x : x ∈ R and x ≠ n π, n ∈ Z} and range is the set {y : y ∈ R, y ≥ 1 or y ≤ – 1}. Similarly, the domain of y = sec x is the set {x : x ∈ R and x ≠ (2n + 1)
π/2, n ∈ Z} and range is the set {y : y ∈ R, y ≤ – 1 or y ≥ 1}. The domain of y = tan x is the set {x : x ∈ R and x ≠ (2n + 1)π/2 , n ∈ Z} and range is the set of all real numbers. The domain of y = cot x is the set {x : x ∈ R and x ≠ n π, n ∈ Z} and the range is the set of all real numbers.
We further observe that in the first quadrant, as x increases from 0 to π/2 , sin x increases from 0 to 1, as x increases from π/2 to π, sin x decreases from 1 to 0. In the third quadrant, as x increases from π to 3π/2, sin x decreases from 0 to –1 and finally, in the fourth quadrant, sin x increases from –1 to 0 as x increases from 3π/2 to 2π.

Similarly, we can discuss the behaviour of other trigonometric functions. In fact, we have the following table:
Remark In the above table, the statement tan x increases from 0 to ∞(infinity) for 0 < x < π/2. Simply means that tan x increases as x increases for 0 < x < π/2 and assumes arbitraily large positive values as x approaches to π/2 . Similarly, to say that cosec x decreases from –1 to –∞(minus infinity) in the fourth quadrant means that cosec x decreases for x ∈ (3π/2, 2π) and assumes arbitrarily large negative values as x approaches to 2π. The symbols ∞ and – ∞ simply specify certain types of behavior of functions and variables.
We have already seen that values of sin x and cos x repeats after an interval of 2π. Hence, values of cosec x and sec x will also repeat after an interval of 2π. We shall see in the next section that tan (π + x) = tan x. Hence, values of tan x will repeat after an interval of π. Since cot x is reciprocal of tan x, its values will also repeat after an interval of π. Using this knowledge and behaviour of trigonometic functions, we can sketch the graph of these functions. The graph of these functions are given below:

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